Optimal. Leaf size=31 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} 2^x}{\sqrt{a+b 4^x}}\right )}{\sqrt{b} \log (2)} \]
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Rubi [A] time = 0.0375751, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2249, 217, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} 2^x}{\sqrt{a+b 4^x}}\right )}{\sqrt{b} \log (2)} \]
Antiderivative was successfully verified.
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Rule 2249
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{2^x}{\sqrt{a+2^{2 x} b}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,2^x\right )}{\log (2)}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{2^x}{\sqrt{a+4^x b}}\right )}{\log (2)}\\ &=\frac{\tanh ^{-1}\left (\frac{2^x \sqrt{b}}{\sqrt{a+4^x b}}\right )}{\sqrt{b} \log (2)}\\ \end{align*}
Mathematica [A] time = 0.0043184, size = 33, normalized size = 1.06 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} 2^x}{\sqrt{a+b 2^{2 x}}}\right )}{\sqrt{b} \log (2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{{2}^{x}{\frac{1}{\sqrt{a+{2}^{2\,x}b}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62163, size = 198, normalized size = 6.39 \begin{align*} \left [\frac{\log \left (-2 \, \sqrt{2^{2 \, x} b + a} 2^{x} \sqrt{b} - 2 \cdot 2^{2 \, x} b - a\right )}{2 \, \sqrt{b} \log \left (2\right )}, -\frac{\sqrt{-b} \arctan \left (\frac{2^{x} \sqrt{-b}}{\sqrt{2^{2 \, x} b + a}}\right )}{b \log \left (2\right )}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.823697, size = 85, normalized size = 2.74 \begin{align*} \frac{\begin{cases} \frac{\sqrt{- \frac{a}{b}} \operatorname{asin}{\left (2^{x} \sqrt{- \frac{b}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge b < 0 \\\frac{\sqrt{\frac{a}{b}} \operatorname{asinh}{\left (2^{x} \sqrt{\frac{b}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge b > 0 \\\frac{\sqrt{- \frac{a}{b}} \operatorname{acosh}{\left (2^{x} \sqrt{- \frac{b}{a}} \right )}}{\sqrt{- a}} & \text{for}\: b > 0 \wedge a < 0 \end{cases}}{\log{\left (2 \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24596, size = 42, normalized size = 1.35 \begin{align*} -\frac{\log \left ({\left | -2^{x} \sqrt{b} + \sqrt{2^{2 \, x} b + a} \right |}\right )}{\sqrt{b} \log \left (2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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